.file "erf.s"// Copyright (c) 2001 - 2005, Intel Corporation// All rights reserved.//// Contributed 2001 by the Intel Numerics Group, Intel Corporation//// Redistribution and use in source and binary forms, with or without// modification, are permitted provided that the following conditions are// met://// * Redistributions of source code must retain the above copyright// notice, this list of conditions and the following disclaimer.//// * Redistributions in binary form must reproduce the above copyright// notice, this list of conditions and the following disclaimer in the// documentation and/or other materials provided with the distribution.//// * The name of Intel Corporation may not be used to endorse or promote// products derived from this software without specific prior written// permission.// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS // "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT // LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL INTEL OR ITS // CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,// EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, // PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR // PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY // OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY OR TORT (INCLUDING// NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS // SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. // // Intel Corporation is the author of this code, and requests that all// problem reports or change requests be submitted to it directly at // http://www.intel.com/software/products/opensource/libraries/num.htm.//// History//==============================================================// 08/15/01 Initial version// 05/20/02 Cleaned up namespace and sf0 syntax// 02/06/03 Reordered header: .section, .global, .proc, .align// 03/31/05 Reformatted delimiters between data tables//// API//==============================================================// double erf(double)//// Overview of operation//==============================================================// Background////// There are 9 paths:// 1. x = +/-0.0//    Return erf(x) = +/-0.0//// 2. 0.0 < |x| < 0.5//    Return erf(x) = x *Pol9(x^2)//// 3. For several subranges of 0.5 <= |x| < 5.90625//    Return erf(x) = sign(x)*Pol19(y), //    where y = (|x|-b)/a, Pol19(y) = A0 + A1*y^1 + A2*y^2 + ... + A19*y^19////    For each subrange there is particular set of coefficients.//    Below is the list of subranges://    3.1 0.5 <= |x| < 1.0     b = a = 0.5//    3.2 1.0 <= |x| < 2.0,    b = a = 1.0//    3.3 2.0 <= |x| < 3.25    b = a = 2.0//    3.4 4.0 <= |x| < 5.90625 b = 4.0, a = 2.0//// 4. 3.25 <= |x| < 4.0//    Return erf(x) = sign(x)*Pol14(|x| - 3.25)//// 5. 5.90625 <= |x| < +INF//    Return erf(x) = sign(x)*(1.0d - 2^(-63))//// 6. |x| = INF//    Return erf(x) = sign(x) * 1.0//// 7. x = [S,Q]NaN //    Return erf(x) = QNaN//// 8. x is positive denormal//    Return erf(x) = A0*x - x^2,//    where A0 = 2.0/sqrt(Pi)//// 9. x is negative denormal//    Return erf(x) = A0*x + x^2,//    where A0 = 2.0/sqrt(Pi)//// Registers used//==============================================================// Floating Point registers used: // f8, input, output// f32 -> f63// General registers used:  // r32 -> r48, r2, r3// Predicate registers used:// p0, p6 -> p15// p6           to filter out case when x = denormal// p7           to filter out case when x = [Q,S]NaN or +/-0,//              used also to process denormals// p8           to filter out case when 3.25 <= |x| < 4.0, //              used also to process denormals// p9           to filter out case when |x| = inf// p10          to filter out case when |x| < 0.5// p11          set when |x| < 3.25 or |x| > 4.0// p12          to filter out case when |x| >= 5.90625// p13          set if 4.0 <=|x| < 5.90625// p14          set to 1 for positive x// p15          set to 1 for negative x// Assembly macros//==============================================================rDataPtr           = r2rDataPtr1          = r3rBias              = r33rCoeffAddr3        = r34rThreeAndQ         = r35rCoeffAddr2        = r36rMask              = r37rArg               = r38rSignBit           = r39rAbsArg            = r40rSaturation        = r41rIndex             = r42rCoeffAddr1        = r43rCoeffAddr4        = r44rShiftedArg        = r45rShiftedArgMasked  = r46rBiasedExpOf4      = r47rShiftedAbsArg     = r48//==============================================================fA0                = f32fA1                = f33fA2                = f34fA3                = f35fA4                = f36fA5                = f37fA6                = f38fA7                = f39fA8                = f40fA9                = f41fA10               = f42fA11               = f43fA12               = f44fA13               = f45fA14               = f46fA15               = f47fA16               = f48fA17               = f49fA18               = f50fA19               = f51fArgSqr            = f52fArgAbsNorm        = f53fSignumX           = f54fRes               = f55fThreeAndQ         = f56fArgAbs            = f57fTSqr              = f58fTQuadr            = f59fTDeg3             = f60fTDeg7             = f61fArgAbsNormSgn     = f62                          fTQuadrSgn         = f63// Data tables//==============================================================RODATA.align 64LOCAL_OBJECT_START(erf_data)// Coefficients ##0..15// Polynomial coefficients for the erf(x), 0.5 <= |x| < 1.0 data8 0xB69AC40646D1F6C1, 0x00003FD2 //A19data8 0x90AD48C0118FA10C, 0x00003FD7 //A18data8 0x826FBAD055EA4AB8, 0x0000BFDB //A17data8 0x8DAB171246CC2B89, 0x00003FDC //A16data8 0xC0B1D6662F8A7564, 0x00003FDF //A15data8 0xA46374AC35099BAF, 0x0000BFE1 //A14data8 0xB2F230996346EF27, 0x0000BFE4 //A13data8 0xCDEC50950FACE04A, 0x00003FE6 //A12data8 0x826014649396E9D2, 0x00003FE9 //A11data8 0xCDB787DC718B13F9, 0x0000BFEB //A10data8 0x8E0B23C24EE0C8EE, 0x0000BFED //A9data8 0xA49EA40A4E5A3F76, 0x00003FF0 //A8data8 0xB11E30BE912617D3, 0x00003FF0 //A7data8 0xCCF89D9351CE26E3, 0x0000BFF4 //A6data8 0xEFF75AD1F0F22809, 0x00003FF2 //A5data8 0xBB793EF404C09A22, 0x00003FF8 //A4// Polynomial coefficients for the erf(x), 1.0 <= |x| < 2.0 data8 0xBAE93FF4174EA59B, 0x00003FE6 //A19data8 0x8A0FD46092F95D44, 0x0000BFEA //A18data8 0xA37B3242B7809E12, 0x00003FEC //A17data8 0xA0330A5CD2E91689, 0x0000BFED //A16data8 0x8E34A678F3497D17, 0x0000BFEC //A15data8 0xAC185D45A2772384, 0x00003FEF //A14data8 0xB0C11347CE7EEDE8, 0x00003FEF //A13data8 0xD3330DC14EA0E4EB, 0x0000BFF2 //A12data8 0xB4A6DFDE578A428F, 0x00003FF1 //A11data8 0xA0B4034310D2D9CB, 0x00003FF5 //A10data8 0xF71662D3132B7759, 0x0000BFF5 //A9data8 0x9C88BF157695E9EC, 0x0000BFF7 //A8data8 0xF84B80EFCA43895D, 0x00003FF8 //A7data8 0x9722D22DA628A17B, 0x00003FF7 //A6data8 0x8DB0A586F8F3381F, 0x0000BFFB //A5data8 0x8DB0A5879F87E5BE, 0x00003FFB //A4// Polynomial coefficients for the erf(x), 2.0 <= |x| < 3.25 data8 0x9C4AF1F3A4B21AFC, 0x00003FF6 //A19data8 0x8D40D5D5DB741AB8, 0x0000BFF9 //A18data8 0xDEBE7099E0A75BA4, 0x00003FFA //A17data8 0xB99A33294D32429D, 0x0000BFFB //A16data8 0x8109D9C7197BC7C9, 0x00003FFB //A15data8 0xC30DE8E2EFC2D760, 0x00003FFA //A14data8 0x80DDA28C5B35DC73, 0x0000BFFC //A13data8 0x9BE4DE5095BACE0D, 0x00003FF9 //A12data8 0xDA4092509EE7D111, 0x00003FFC //A11data8 0x89D98C561B0C9040, 0x0000BFFD //A10data8 0xD20B26EB2F0881D4, 0x0000BFF9 //A9data8 0xD089C56948731561, 0x00003FFD //A8data8 0xDD704DEFFB21B7E7, 0x0000BFFD //A7data8 0xF0C9A6BBDE469115, 0x00003FF9 //A6data8 0xD673A02CB5766633, 0x00003FFD //A5data8 0x8D162CBAD8A12649, 0x0000BFFE //A4// Polynomial coefficients for the erf(x), 4.0 <= |x| < 6.0  data8 0xD4428B75C6FE8FD1, 0x0000BFFC //A19data8 0xF76BE1935675D5C8, 0x00003FFE //A18data8 0xFD6BB3B14AA7A8E6, 0x0000BFFF //A17data8 0x8BE8F573D348DDA4, 0x00004000 //A16data8 0x81E91923A1030502, 0x0000BFFF //A15data8 0xCE7FE87B26CFD286, 0x0000BFFE //A14data8 0x84EF6B4E17404384, 0x00004000 //A13data8 0x91FEF33015404991, 0x0000C000 //A12data8 0xDEDF6A9370747E56, 0x00003FFF //A11data8 0x8397E6FF56CDFD9D, 0x0000BFFF //A10data8 0xFAD1CE912473937B, 0x00003FFD //A9data8 0xC48C1EA8AAA624EA, 0x0000BFFC //A8data8 0xFECAF0097ACF981B, 0x00003FFA //A7data8 0x8829A394065E4B95, 0x0000BFF9 //A6data8 0xED3003E477A53EE7, 0x00003FF6 //A5data8 0xA4C07E9BB3FCB0F3, 0x0000BFF4 //A4//// Coefficients ##16..19// Polynomial coefficients for the erf(x), 0.5 <= |x| < 1.0 data8 0x95FA98C337005D13, 0x0000BFF9 //A3data8 0xE0F7E524D2808A97, 0x0000BFFB //A2data8 0xE0F7E524D2808A98, 0x00003FFD //A1data8 0x853F7AE0C76E915F, 0x00003FFE //A0// Polynomial coefficients for the erf(x), 1.0 <= |x| < 2.0 data8 0x8DB0A587A96ABCF0, 0x00003FFC //A3data8 0xD488F84B7DE18DA8, 0x0000BFFD //A2data8 0xD488F84B7DE12E9C, 0x00003FFD //A1data8 0xD7BB3D3A08445636, 0x00003FFE //A0// Polynomial coefficients for the erf(x), 2.0 <= |x| < 3.25data8 0xC58571D23D5C4B3A, 0x00003FFD //A3data8 0xA94DCF467CD6AFF3, 0x0000BFFC //A2data8 0xA94DCF467CD10A16, 0x00003FFA //A1data8 0xFECD70A13CAF1997, 0x00003FFE //A0 // Polynomial coefficients for the erf(x), 4.0 <= |x| < 6.0 data8 0xB01D2B4F0D5AB8B0, 0x00003FF1 //A3data8 0x8858A465CE594BD1, 0x0000BFEE //A2data8 0x8858A447456DE61D, 0x00003FEA //A1data8 0xFFFFFFBDC88BB107, 0x00003FFE //A0// Polynomial coefficients for the erf(x), 0.0 <= |x| < 0.5 data8 0xBE839EDBB36C7FCE //A9data8 0x3EBB7745A18DD242 //A8data8 0xBF4C02DB238F2AFC //A5data8 0x3F7565BCD0A9A3EA //A4data8 0xC093A3581BCF3333, 0x0000BFFD //A1data8 0xBEEF4BB82AD8AE22 //A7data8 0x3F1F9A2A57A218CD //A6data8 0xBF9B82CE3127F4E4 //A3data8 0x3FBCE2F21A042B25 //A2data8 0x906EBA8214DB688D, 0x00003FFF //A0// 1.0 - 2^(-63)data8 0xFFFFFFFFFFFFFFFF, 0x00003FFE // Polynomial coefficients for the erf(x), 3.25 <= |x| < 4.0 data8 0x95E91576C7A12250, 0x00003FE7 //A14data8 0x8E5E0D0E1F5D3CB5, 0x0000BFEA //A13data8 0xED761DAFAF814DE9, 0x00003FEB //A12data8 0xB3A77D921D0ACFC7, 0x0000BFEC //A11data8 0xA662D27096B08D7C, 0x0000BFEC //A10data8 0xDA0F410AE6233EA5, 0x00003FEF //A9data8 0xAB4A8B16B3124327, 0x0000BFF1 //A8data8 0xB241E236A5EDCED3, 0x00003FF2 //A7data8 0x8A2A65BA1F551F77, 0x0000BFF3 //A6data8 0xA4852D0B1D87000A, 0x00003FF3 //A5data8 0x963EB00039489476, 0x0000BFF3 //A4data8 0xCD5244FF4F7313A5, 0x00003FF2 //A3data8 0xC6F1E695363BCB26, 0x0000BFF1 //A2data8 0xF4DAF4680DA54C02, 0x00003FEF //A1data8 0xFFFFB7CFB3F2ABBE, 0x00003FFE //A0// A = 2.0/sqrt(Pi)data8 0x906EBA8214DB688D, 0x00003FFF LOCAL_OBJECT_END(erf_data).section .textGLOBAL_LIBM_ENTRY(erf){ .mfi      alloc          r32 = ar.pfs, 0, 17, 0, 0      fmerge.se      fArgAbsNorm = f1, f8         // normalized x      adds           rSignBit = 0x1, r0}{ .mfi      addl           rDataPtr = @ltoff(erf_data), gp      fma.s1         fArgSqr = f8, f8, f0         // x^2      addl           rThreeAndQ = 0x400A0, r0     // shifted bits of 3.25};;{ .mfi      getf.d         rArg = f8                    // x in GR       fclass.m       p6,p0 = f8, 0x0b             // is x denormal ?      shl            rThreeAndQ = rThreeAndQ, 44  // bits of 3.25}{ .mfi      ld8            rDataPtr = [rDataPtr]      nop.f          0      addl           rBiasedExpOf4 = 0x40100, r0  // shifted bits of 4.0};;{ .mfi      addl           rSaturation = 0x4017A, r0    // shifted bits of 5.90625      fclass.m       p7,p0 = f8, 0xc7             // is x [S,Q]NaN or +/-0 ?      shl            rSignBit = rSignBit, 63      // mask for sign bit}{ .mfi      addl           rMask = 0x7FF00, r0          // Mask for index bits      nop.f          0      addl           rBias = 0x3FE00, r0          // bias of 0.5 << 8};;{ .mfi      setf.d         fThreeAndQ = rThreeAndQ      // 3.25 if FP register      fclass.m       p9,p0 = f8, 0x23             // is x +/- inf?      shr.u          rShiftedArg = rArg, 44}{ .mfb      andcm          rAbsArg = rArg, rSignBit     // |x| in GR      nop.f          0(p6)  br.cond.spnt   erf_denormal                 // branch out if x is denormal};;   { .mfi      and            rShiftedArgMasked = rShiftedArg, rMask // bias of x << 8      fmerge.s       fArgAbs = f1, f8             // |x|      shr            rShiftedAbsArg = rAbsArg, 44}{ .mfb      cmp.lt         p8, p11 = rThreeAndQ, rAbsArg // p8 = 1 if |x| >= 3.25(p7)  fma.d.s0       f8 = f8,f1,f8                // NaN or +/-0(p7)  br.ret.spnt    b0                           // exit for x = NaN or +/-0};;              { .mfi      sub            rIndex = rShiftedArgMasked, rBias // index << 8      nop.f          0       cmp.lt         p10, p0 = rShiftedArgMasked, rBias // p10 = 1 if |x| < 0.5 }{ .mfb      // p8 = 1 if 3.25 <= |x| < 4.0 (p8)  cmp.lt         p8, p11 = rShiftedAbsArg, rBiasedExpOf4       fms.s1         fArgAbsNorm = fArgAbsNorm, f1, f1(p10) br.cond.spnt   erf_near_zero // branch out if |x| < 0.5};;.pred.rel "mutex", p8, p11{ .mfi(p8)  adds           rCoeffAddr1 = 1392, rDataPtr // coeff. for 3.25 <=|x|<4.0(p9)  fmerge.s       f8 = f8,f1                   // +/- inf      nop.i          0}{ .mfb(p11) add            rCoeffAddr1 = rDataPtr, rIndex// coeff. ##0,2,..14      nop.f          0(p9)  br.ret.spnt    b0                            // exit for x = +/- inf};;{ .mfi      adds           rCoeffAddr2 = 16, rCoeffAddr1       fmerge.s       fSignumX = f8, f1            // signum(x)      nop.i          0} { .mfb      cmp.lt         p12, p0 = rSaturation, rShiftedAbsArg // |x| > 5.90625?      nop.f          0(p12) br.cond.spnt   erf_saturation               // branch out if x |x| >= 6.0};;// Here if paths #3,4// if path #4 we'll branch out after loading of 14 necessary coefficients{.mfi      ldfe           fA19 = [rCoeffAddr1], 32      nop.f          0      nop.i          0}{.mfi      ldfe           fA18 = [rCoeffAddr2], 32      nop.f          0      adds           rCoeffAddr3 = 1024, rDataPtr};;{.mfi      ldfe           fA17 = [rCoeffAddr1], 32      nop.f          0      nop.i          0}{.mfi      ldfe           fA16 = [rCoeffAddr2], 32      nop.f          0      nop.i          0};;{.mfi      ldfe           fA15 = [rCoeffAddr1], 32      fma.s1         fTSqr = fArgAbsNorm, fArgAbsNorm, f0      shr.u          rIndex = rIndex, 2}{.mfi      ldfe           fA14 = [rCoeffAddr2], 32      nop.f          0      adds           rCoeffAddr4 = 16, r0};;{.mfi      ldfe           fA13 = [rCoeffAddr1], 32      nop.f          0      // address of coefficients ##16..23      add            rCoeffAddr3 = rCoeffAddr3, rIndex }{.mfi      ldfe           fA12 = [rCoeffAddr2], 32      nop.f          0      cmp.lt         p15, p14 = rArg, r0};;{.mfi      ldfe           fA11 = [rCoeffAddr1], 32      nop.f          0      add            rCoeffAddr4 = rCoeffAddr3, rCoeffAddr4}{.mfi      ldfe           fA10 = [rCoeffAddr2], 32      nop.f          0      nop.i          0};;{.mfi      ldfe           fA9 = [rCoeffAddr1], 32      nop.f          0      nop.i          0}{.mfi      ldfe           fA8 = [rCoeffAddr2], 32